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AS // PQ wanted, starting with 2 intersecting circles

Source: VI Soros Olympiad 1990-00 R3 10.4 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

May 28, 2024
geometryparallel

Problem Statement

The circles ω1\omega_1 and ω2\omega_2 intersect at two points AA and BB. On the circle ω2\omega_2, point CC is taken in such a way that CACA is tangent to the circle ω1\omega_1. Through point AA, a straight line is drawn that intersects the circles ω1\omega_1, and ω2\omega_2 at points MM and NN, respectively , different from point AA. Point PP is the midpoint of the segment ACAC, QQ is the midpoint of MNMN, and SS is the intersection point of the line BQBQ with the circle ω1\omega_1, different from point BB. Prove that the lines ASAS and PQPQ are parallel.
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